Biomedical Engineering Reference
In-Depth Information
The price paid for a reduction in leakage and scalloping through the use of a nonrec-
tangular window is reduced resolution. In fact, if it is necessary to view two closely spaced
peaks, the rectangular window's narrow main lobe will allow the user to obtain analysis
results that report the existence of these closely spaced components, whereas any of the
other windows would end up fusing these two peaks into a single smooth crest.
Use of a rectangular window is also appropriate for the analysis of transients. In these
cases, zero signal usually precedes and succeeds the transient. Thus, if the FFT is made to
look at the complete data record for the transient, no arti
cial discontinuities are intro-
duced, and full resolution can be obtained without leakage. As you may well see, there is
no single window that outperforms all others in every respect, and it is safe to say that
selecting the appropriate window for a speci
fi
fi
c application is more of an art than an exact
science.
When the signal rides on a relatively high dc level or on a strong sinusoidal signal, it is
advisable to remove these components from the data before the PSD is estimated, because
otherwise the biasing and strong sidelobes produced by them could easily obscure weaker
components. Whenever expected physically, the dc component of a signal can usually be
removed by subtracting the sampled data mean
x
N
1
(1/
N
)
x
n
from each data sample to
n
0
produce a “purely ac” data sequence
x
0
x
,
x
1
x
,
...
,
x
N
1
x
.
Zero-Padding an FFT
An interesting property of the FFT is that simply adding zeros after a windowed data sam-
ple sequence
x
0
,
x
1
,
...
,
x
N
1
in order to create a longer record
x
0
,
x
1
,
...
,
x
N
1
,0,0,...,0
before performing an FFT will cause the FFT to interpolate transform values between the
N
original transform values. This process, called
zero padding
, is often mistakenly thought of
as a trick to improve the inherent resolution of an FFT. Zero padding will also provide a
much smoother PSD and will help annul ambiguities regarding the power and location of
peaks that may be scalloped by the non-zero-padded FFT.
Classical Methods
As mentioned before, a common mistake is to assume that the solution to
P
(
f
m
)
1, the
periodogram
, is a reliable estimate of the PSD. Actual
proof of this is beyond the scope of the topic, but it has been demonstrated that regardless
of how large
N
(the number of available data samples) is, the statistical variance of the esti-
mated periodogram spectrum does not tend to zero. This statistical inconsistency is respon-
sible for the lack of reliability of the periodogram as a spectral estimator.
The solution to this problem is simple, however. If a number of periodograms are com-
puted for di
X
m
2
/
N
∆
t
,
m
0, 1,
...
,
N
erent segments of a data record, their average results in a PSD estimate with
good statistical consistency. Based on this, Welch [1967] proposed a simple method to
determine the average of a number of periodograms computed from overlapping segments
of the data record available. Welch's PSD estimate
P
^
(
f
) of
M
data samples is the average
of
K
periodograms
P
(
f
) of
N
points each:
ff
K
1
P
^
(
f
)
K
P
(
f
)
i
1
where the
P
(
f
)'s are obtained by applying
P
(
f
m
)
1to
appropriately weighted data. It is obvious that if the original
M
-point data record is divided
into segments of
N
points each, with a shift of
S
samples between adjacent segments, the
number of periodograms that can be averaged is
X
m
2
/
N
∆
t
,
m
0, 1,
...
,
N
S
1
N
K
integer
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